1 / 18

Game Theory Dynamic Bayesian Games

Game Theory Dynamic Bayesian Games. Univ. Prof.dr. M.C.W. Janssen University of Vienna Winter semester 2011-12 Week 2 (January 9-10). Difference Dynamics and Statics.

ordell
Download Presentation

Game Theory Dynamic Bayesian Games

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Game Theory Dynamic Bayesian Games Univ. Prof.dr. M.C.W. Janssen University of Vienna Winter semester 2011-12 Week 2 (January 9-10)

  2. Difference Dynamics and Statics • The only thing to learn in static games with asymmetric information is when types are correlated and then information about own type reveals info about types of other players • Usually, independent types are assumed • In dynamic games with asymmetric information players may learn about types of other players through actions that are chosen before they themselves have to make decisions

  3. Important class of signaling games • In signaling games there are two players, Sender and Receiver • Type of Sender is private information, sender takes an action • Strategy is action depending on type • Receiver takes an action after observing action taken by the sender • Type of sender may be inferred (revealed) on the basis of the action that is actually taken

  4. A simple example 3,2 accept reject U -1,0 apply S 0,0 1/2 N.A good -1,-3 N accept 1/2 apply reject U -1,0 bad S N.A 0,0

  5. What is an equilibrium in this example? • Strategies for both players such that both players strategies are optimal given strategy of the other and the possibly updated beliefs • Pooling and separating equilibria possible (as well as –in more complicated games- semi-pooling or semi-separating equilibria) • Pooling equilibria are equilibria where all types of the sender choose the same action • (Fully) Separating equilibria are equilibria where all types choose different actions from one another (space of actions and types should allow this)

  6. A Separating equilibrium • Proposal: University accepts, student applies if, and only if she is good • Check: Can a player benefit by deviating? • Good student gets positive pay-off in equilibrium, if she deviates she gets 0; • Bad student does not want to apply as this would give a negative pay-off of -1 instead of the equilibrium pay-off of 0; • University can get a pay-off of 2 or -3 when accepting. How to weigh these pay-offs? Bayes’ Rule says that P(good student|application) = 1. Thus, equilibrium pay-off is 2 and deviating gives lower pay-off (of 0) • Thus, this is a separating equilibrium • Are there other separating equilibria?

  7. A Pooling equilibria • Proposal: University rejects, student never applies • Check: Can one of the players benefit by deviating • University always gets pay-off of 0 if student does not apply. Deviating does not improve his situation. • Student does not want to apply (whatever her type) knowing she will be rejected; pay-off of -1 instead of the equilibrium pay-off of 0) • Thus, this is a pooling equilibrium • Are there other pooling equilibria?

  8. The example changed 3,2 accept reject U -1,0 apply S 0,0 1/2 N.A good 1,1 N accept 1/2 apply reject U -1,0 bad S N.A 0,0

  9. Is pooling equilibrium reasonable? • First, it is important to realize that pooling on not applying is still part of an equilibrium • But, in this modified game, it does not seem reasonable. Why? • Rejecting students always gives a pay-off of 0, whereas accepting gives a positive pay-off • Thus, rejecting is an incredible threat • How to get rid of incredible threats? (Usually, impose subgame perfection. But how many subgames are there?)

  10. Perfect Bayes-Nash equilibrium • Refinement of Bayes-Nash equilibrium • Often, in games with private information some information sets are off-the-equilibrium path • When an information set is off-the-equilibrium path, Bayes’ Rule cannot be applied (gives 0/0) • Bayes-Nash equilibrium does not impose any restrictions on strategy after such info set • Perfect Bayes-Nash equilibrium says that (i) some out-of-equilibrium beliefs have to be specified and (ii) given these beliefs, actions have to be optimal

  11. Definition PBNE • A Perfect Bayes-Nash equilibrium is a set of strategies s, one for each player, and out-of-equilibrium beliefs μ(.|a) such that i. each player chooses an optimal strategy given strategies of other players evaluated at updated beliefs iia. μ(.|a) is formed using Bayes’ Rule whenever possible, i.e., if ∑θ p(θ)σ(a|θ) > 0 iib. μ(.|a) is any (arbitrary) probability distribution over type space Θ if ∑θ p(θ)σ(a|θ) = 0

  12. Ruling out pooling equilibrium in modified example • PBNE requires P(good student|apply) to be specified. Let’s say it is μ, where 0≤ μ ≤1 • Thus, P(bad student|apply) = 1 – μ • The expected pay-off for the University of accepting a student is then 2μ + (1-μ), which is positive for any permissible value of μ. • Therefore, University cannot reject a student if it receives an application, as this is not optimal given any out-of-equilibrium belief μ. • With separating equilibrium, one does not need to specify out-of-equilibrium beliefs, as no information set is out-of-equilibrium.

  13. Sequential Rationality • Sequential rationality extends and refines PBNE • Rationality condition of PBNE is extended to hold for any info set • Beliefs condition is refined to include info sets that do not arise from asymmetric information

  14. Example where sequential rationality is needed • Two NE: (T,U) and (B,D) • SPE and PBNE do not have any bite • Still (T,U) seems unreasonable as it is always best for player 2 to choose D • But SPE and PBNE do not require to specify beliefs in this situation 2,2 T 0,0 U M D 0,1 B 1,0 U D 3,1

  15. Reconsider simple example 3,2 accept reject U -1,0 apply S 0,0 1/2 N.A good -1,-3 N accept 1/2 apply reject U -1,0 bad S N.A 0,0 15

  16. Is pooling reasonable? • It satisfies PBNE • Specify out-of-eq beliefs P(good|apply) < 3/5 • It satisfies sequential equilibrium • Sequence σn where both types of students choose to apply with probability 1/n. Updating beliefs gives P(good|apply) = ½ so that university has to reject along the sequence σn • But still it seems that the out-of-equilibrium belief should be P(good|apply) = 1. Why? To apply is a dominated strategy for bad student. (He has never incentive to apply; whereas good student may have incentive if she is accepted) • How to formalize this?

  17. Domination-based beliefs I • Action a is strictly dominated for type θ if there is another action a’ s.t. Mins’S u(a’,s’,θ) > maxsS u(a,s,θ) • For each action aA define Θ(a) = {θ: there is no a’A that strictly dominates a} • A PBE has reasonable beliefs if for all aA with Θ(a) ≠ empty set, μ(θ / a) > 0 only if θΘ(a) • Apply this definition to simple example

  18. Intermediate step to Intuitive Criterion 3,2 accept reject U -1,0 apply exam S 2,1 0,0 1/2 N.A good -1,-3 N accept 1/2 apply reject U -1,0 bad exam S 1,-4 N.A 0,0 18

More Related